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I can understand the adjugate matrix and the motivation of that to find the inverse, but I can't see how this idea was invented by mathematicians. It's just brilliance or someone understand how the properties were conceived.

I trying to understand this property origins, but I can't explain how cofactor were created too:

$$\sum_{i = 1}^n a_{ij} C_{ik}= \begin{cases} \det(A) & \text{if } j = k \\ 0 & \text{if } j \neq k \end{cases}$$

Allen Knutson
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Ráfagan
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    Try to look at the inverse matrix by finding the indeterminate entries with the Cramer's rule (which is rather old mathematics), and the matrix of cofactors appears quite naturally. That said, this is not a research question, so I vote to close. – Francesco Polizzi Oct 07 '15 at 08:55
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    Allen Knutson parked his rephrasing of the question together with an answer at https://plus.google.com/+AllenKnutson/posts/LgLxgxsXNAT. – Christian Stump Oct 07 '15 at 12:19
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    I vote to reopen: http://meta.mathoverflow.net/a/2519/1 – Anton Geraschenko Oct 07 '15 at 18:01
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    I think the Wikipedia entry of Cramer's rule is quite accessible and well motivated (and relevant here). Also, Thomas Muir posted a history of determinants (which existed before matrix notation!) and their development. You can find both resources on the web. Also, the question could use an example of what shape is wanted for the answer: pointers to the literature, a copy of the Wikipedia article, a category theory or foundational approach, or something understood by someone with only one or two linear algebra courses behind them. Gerhard "Falls Into The Last Category" Paseman, 2015.10.07 – Gerhard Paseman Oct 07 '15 at 19:08
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    http://mathoverflow.net/a/89079/290 – Qiaochu Yuan Oct 07 '15 at 19:42
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    (My answer linked above has no essential difference from Qiaochu's answer, linked directly above.) – Allen Knutson Oct 07 '15 at 21:26
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    "Thomas Muir posted a history of determinants...." Posted? Really, @Gerhard? – Gerry Myerson Oct 07 '15 at 21:57
  • François Dorais also posted an answer here: https://plus.google.com/+FrancoisDorais/posts/C6SBgvj5v8e – David Roberts Oct 08 '15 at 03:02
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    Indeed. He reworked his research several times, with posts in 1890 and 1906. A copy of one of them can be found here: https://archive.org/details/theoryofdetermin01muiruoft . People were saying to him "Inter-net? Whereof speakest thou?", to which Muir calmly replied "Just wait." Gerhard "Clearly Ahead Of His Time" Paseman, 2015.10.08 – Gerhard Paseman Oct 08 '15 at 15:43

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